Quasianalytic Banach function algebras

Dales, H.G. and Davie, A. M. (1973) Quasianalytic Banach function algebras. Journal of Functional Analysis, 13 (1). pp. 28-50. ISSN 0022-1236

Abstract

We construct certain Banach algebras of infinitely differentiable functions on compact plane sets such that the algebras are quasianalytic, and we use these algebras to construct examples of Banach algebras defined on their maximal ideal spaces which, first, have only countably many peak points and, second, have the property that a discontinuous function operates on the algebra. We show that any function defined on an open subset of the plane which operates on a Banach function algebra is necessarily continuous on a dense open subset of its domain.